Lines and Planes, Part 2

Lines and Planes in Three Dimensions

Part 2: Planes in Three Dimensional Space

Planes in space can be defined using vectors as well. In fact, when defining the cross product of two vectors, we speak of the resulting vector as being perpendicular to the plane in which the original two vectors lie.

Turning this idea around, we define a plane to be the set of all vectors perpendicular to a given vector. As a result of this definition, the dot product will be of use describing the equations that yield a plane in space.

Computationally, we can quickly give an equation that characterized a plane. This is done by computing the dot product of a vector n = (a,b,c) with an arbitrary vector (x,y,z).

Find the equation of a plane perpendicular to a vector n by computing this dot product.
Find the equation of the plane perpendicular to (-1,1,-1).
What do you notice about the planes perpendicular to the coordinate axes. Compute the equation of the plane orthogonal to the vector (0,0,1).
Lines in the cartesian plane have degree one. Explain why it makes sense to think of planes in three dimensional space as the analogue of lines in the two dimensional cartesian plane.

From step 1, we find that the equation of a plane is ax + by + cz = 0 but this is not the most general form. Planes of this type intersect the origin in space. In general, a plane may have equation

ax + by + cz = d

where d can have any value. When this occurs, the plane is being translated away from the origin by some vector r₀ as depicted in the diagram below.

Show that, effectively, d is either 0 or 1. That is, show that any other case reduces to these two.
Explain why the case where d = 0 means that the plane must intersect the origin.

In general, the vector equation for a plane becomes <(r - r₀),n> = 0. Here, n is the normal vector to the plane and r₀ is the vector translating the plane away from the origin.